Evaluate the following: - 2x 1- dx - Vidyadip

Question

Evaluate the following:
$\int\frac{\cos\text{x}-\cos2\text{x}}{1-\cos\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\cos\text{x}-\cos2\text{x}}{1-\cos\text{x}}\text{dx}$
$=\int\frac{2\sin\frac{3\text{x}}{2}\cdot\sin\frac{\text{x}}{2}}{2\sin^2\frac{\text{x}}{2}}\text{dx}$
$=\int\frac{\sin\frac{3\text{x}}{2}}{\sin\frac{\text{x}}{2}}\text{dx}$
$=\int\frac{3\sin\frac{\text{x}}{2}-4\sin^3\frac{\text{x}}{2}}{\sin\frac{\text{x}}{2}}\text{dx}$ $\big[\because\sin3\text{x}=3\sin\text{x}-4\sin^3\text{x}\big]$
$=3\int\text{dx}-4\int\sin^2\frac{\text{x}}{2}\text{dx}$
$=3\int\text{dx}-4\int\frac{1-\cos\text{x}}{2}\text{dx}$ $=\int\text{dx}+2\int\cos\text{xdx}=\text{x}+2\sin\text{x}+\text{C}$

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